Up to 800 visiting fans arrived at the stadium 20 minutes after kick-off and immediately removed the segregation nets on the east corner of the Atyeo Stand.
The situation was initially dealt with by the stewarding team and police in the vicinity.
Twenty-five Swansea City fans were removed during the game and there were a number of arrests.
Regrettably, several Bristol City supporters were involved in altercations in the corner of the Atyeo Stand and Dolman Stand before being apprehended.
As a result, two stewards and a police officer were treated for minor injuries.
Ashton Gate Stadium Managing Director Mark Kelly said: “Ashton Gate Stadium has a very strong record of match day safety and I can assure our fans that any home or away supporters that caused this unpleasant experience will be banned.
“The incident in its entirety was captured on CCTV and a formal investigation in conjunction with the Avon and Somerset Police match commander and our head of safety is under way.
“I would like to take this opportunity to recognise our safety team and stewards for taking control very quickly and dealing with this situation.”